Relative internality and definable fibrations

We first elaborate on the theory of relative internality in stable theories from [13], focusing on the notion of uniform relative internality (called collapse of the groupoid in [13]), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA. We prove that DCF0 does not have the strong canonical base property, correcting a proof in [20]. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper we study definable fibrations in DCF0, where the general fibre is internal to the constants, including differential tangent bundles, and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article first elaborates on the theory of relative internality in stable theories from [13], focusing on the notion of uniform relative internality (called collapse of the groupoid in [13]), and relates it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA. We prove that DCF0 does not have the strong canonical base property, correcting a proof in [20]. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper, we study definable fibrations in DCF0, where the general fiber is internal to the constants, including differential tangent bundles and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants. (c) 2023 Elsevier Inc. All rights reserved.